A339856 Primitive triples for integer-sided triangles whose sides a < b < c form a geometric progression.

4, 6, 9, 9, 12, 16, 16, 20, 25, 25, 30, 36, 25, 35, 49, 25, 40, 64, 36, 42, 49, 49, 56, 64, 49, 63, 81, 49, 70, 100, 49, 77, 121, 64, 72, 81, 64, 88, 121, 81, 90, 100, 81, 99, 121, 81, 117, 169, 81, 126, 196, 100, 110, 121, 100, 130, 169, 121, 132, 144, 121, 143, 169

Offset: 1

Author

Bernard Schott, Dec 19 2020

Keywords

• nonn
• easy
• tabf

Comments

These triangles are called "geometric triangles" in Project Euler problem 370 (see link).

The triples are displayed in increasing lexicographic order (a, b, c).

Equivalently: triples of integer-sided triangles such that b^2 = a*c with a < c and gcd(a, c) = 1.

When a < b < c are in geometric progression with b = a*q, c = b*q, q is the constant, then 1 < q < (1+sqrt(5))/2 = phi = A001622 = 1.6180... (this bound is used in Maple code).

For each triple (a, b, c), there exists (r, s), 0 < r < s such that a = r^2, b = r*s, c = s^2, q = s/r.

Angle C < 90 degrees if 1 < q < sqrt(phi) and angle C > 90 degrees if sqrt(phi) < q < phi with sqrt(phi) = A139339 = 1.2720...

For k >= 2, each triple (a, b, c) of the form (k^2, k*(k+1), (k+1)^2) is (A008133(3k+1), A008133(3k+2), A008133(3k+3)).

Three geometrical properties about these triangles:

1) The sinus satisfy sin^2(B) = sin(A) * sin(C) with sin(A) < sin(B) < sin(C) that form a geometric progression.

2) The heights satisfy h_b^2 = h_a * h_c with h_c < h_b < h_a that form a geometric progression.

3) b^2 = 2 * R * h_b, with R = circumradius of the triangle ABC.

Examples

			The smallest such triangle is (4, 6, 9) with 4*9 = 6^2.
There exist four triangles with small side = 49 corresponding to triples (49, 56, 64), (49, 63, 81), (49, 70, 100) and (49, 77, 121).
The table begins:
   4,  6,  9;
   9, 12, 16;
  16, 20, 25;
  25, 30, 36;
  25, 35, 49;
  25, 40, 64;
  36, 42, 49;
  ...
		

Links

• David A. Corneth, Table of n, a(n) for n = 1..10002
• Project Euler, Problem 370: Geometric Triangles.

Crossrefs

Cf. A339857 (smallest side), A339858 (middle side), A339859 (largest side), A339860 (perimeter).

Cf. A336755 (similar for sides in arithmetic progression).

Cf. A335893 (similar for angles in arithmetic progression).

Cf. A001622 (phi), A139339 (sqrt(phi)), A008133.

Programs

Maple

for a from 1 to 300 do
for b from a+1 to floor((1+sqrt(5))/2 * a) do
for c from b+1 to floor((1+sqrt(5))/2 * b) do
k:=a*c;
if k=b^2 and igcd(a,b,c)=1 then print(a,b,c); end if;
end do;
end do;
end do;

PARI

lista(nn) = {my(phi = (1+sqrt(5))/2); for (a=1, nn, for (b=a+1, floor(a*phi), for (c=b+1, floor(b*phi), if ((a*c == b^2) && (gcd([a,b,c])==1), print([a,b,c])););););} \\ Michel Marcus, Dec 25 2020

PARI

upto(n) = my(res=List(), phi = (sqrt(5)+1) / 2); for(i = 2, sqrtint(n), for(j = i+1, (i*phi)\1, if(gcd(i, j)==1, listput(res, [i^2, i*j, j^2])))); concat(Vec(res)) \\ David A. Corneth, Dec 25 2020

Extensions

Data corrected by David A. Corneth, Dec 25 2020